Spontaneous spin-valley polarization in NbSe2 at a van der Waals interface

A proximity effect at a van der Waals (vdW) interface enables creation of an emergent quantum electronic ground state. Here we demonstrate that an originally superconducting two-dimensional (2D) NbSe2 forms a ferromagnetic ground state with spontaneous spin polarization at a vdW interface with a 2D ferromagnet V5Se8. We investigated the anomalous Hall effect (AHE) of the NbSe2/V5Se8 magnetic vdW heterostructures, and found that the sign of the AHE was reversed as the number of the V5Se8 layer was thinned down to the monolayer limit. Interestingly, the AHE signal of those samples was enhanced with the in-plane magnetic fields, suggesting an additional contribution to the AHE signal other than magnetization. This unusual behavior is well reproduced by band structure calculations, where the emergence of the Berry curvature along the spin-degenerate nodal lines in 2D NbSe2 by the in-plane magnetization plays a key role, unveiling a unique interplay between magnetism and Zeeman-type spin-orbit interaction in a non-centrosymmetric 2D quantum material.


Sample fabrication and characterization.
All the samples were fabricated by MBE by following our growth process 1-3 . The layer number was precisely designed by monitoring the RHEED intensity oscillations during the growth. Supplementary Figures 1a and 1b show typical RHEED intensity oscillations recorded during the growth of the initial V5Se8 layers and the following NbSe2 layers, respectively. The actual layer numbers were confirmed by XRD measurements.
Supplementary Figure 1c shows the out-of-plane XRD pattern of the N = 6 L sample used in this study. The strong diffraction peak was observed at around 14-15° with clear Laue oscillation, indicating high crystalline coherence along the out-of-plane direction. The

The R-T curves of the representative samples.
As we wrote in the main text, we consider that the electrical conductions of the N < 3 L samples in the low temperature regime are governed by the 4 L-thick NbSe2 layer.
Supplementary Figure 2 shows the normalized R-T curves of the individual films and that of the Nb/V heterostructure sample with N = 2.0 L. The V5Se8 individual film showed metallic behavior in the thick-enough regime (30 L), whereas it exhibited weakly insulating behavior in the thin limit (3 L) as we reported in the previous study 1 . On the other hand, the NbSe2 individual film exhibited metallic behavior down to the thin limit

Other calculation results on monolayer NbSe 2 .
In the main text, we discuss the calculation results with the fixed exchange field |M| = 40 meV, but in reality the exchange field is unknown. In this section, we provide the calculation results with different exchange fields, as well as the results on the angle dependence of the AHE at different energies.

(I) The angle dependence of the AHE with different exchange fields.
Supplementary Figure 6a shows the angle dependence of the xy with different |M| at E = EF, and Supplementary Fig. 6b shows the magnitude of a deviation from cos (θ) at θ = 20 o as a function of |M|. The largest deviation is achieved when |M| = 40 meV, corresponding to the situation that the corner of the FS of monolayer NbSe2 near the  valley is contacted with the peak of the emergent Berry curvature surrounding the K valleys originating from the spin-degenerate nodal lines as we discussed in the main text (see Fig. 5f). We however note that such a deviation from cos (θ) could be observed in a rather broad range of the exchange field from a few millielectronvolt to a hundred millielectronvolt (see Supplementary Fig. 6b).

Calculation results on bilayer NbSe 2 .
In the main text, we discuss the band structure of monolayer NbSe2, but in reality our samples have multilayer NbSe2. In this section, we discuss the band structure of bilayer NbSe2 under the exchange field, and demonstrate that a physical picture based on monolayer NbSe2 proposed in the main text could be applicable to bilayer NbSe2 as well.
Multilayer NbSe2 should provide essentially the same results as those of bilayer NbSe2 as long as a proximity effect is limited to one layer in contact with a ferromagnet. are associated with the up-and down-pseudospin, respectively [5][6][7] . We note that the real spins do not contribute to the Berry curvature in this case, as the up-spin band and the down-spin band are not hybridized by the out-of-plane exchange field.
Those band structure and the Berry curvature are largely modulated when the exchange field is tilted to the in-plane direction as shown in Supplementary Fig. 9c. We observe the emergence of the additional Berry curvature as is the case for monolayer NbSe2, part of which should be originating from hybridization of the up-spin band and the down-spin band by the in-plane magnetization. However, there should be another contribution from the orbital pseudospins, which should be also mixed by the in-plane magnetization and generate additional Berry curvature. As a result, a deviation of the xy from cos (θ) becomes much more pronounced in bilayer NbSe2 as will be shown in the next section.

(III) The angle dependence of the AHE of bilayer NbSe 2 .
Supplementary Figure 10a shows the energy dependence of the xy for different θ calculated from the band structure of bilayer NbSe2 shown in Supplementary Fig. 9a. The sign of xy is positive at E = EF as is the case for monolayer NbSe2, which is consistent to the experimental results. Supplementary Figure 10b shows the angle dependences of the xy at E = EF for monolayer and bilayer NbSe2. Clear deviations from cos (θ) are visible for both cases, suggesting that a similar mechanism associated with the emergence and bilayer NbSe2 (red symbols).
of the additional Berry curvature with the in-plane magnetization should be at work for both monolayer and bilayer cases. The details of the behavior are however different, most likely because bilayer NbSe2 has two origins for the additional Berry curvature, the real spins and the orbital pseudospins.